Abstract

The dependence of resonant multiphoton emission of electrons on the relative and carrier-envelope phases of bichromatic laser fields is studied theoretically by numerically solving the time-dependent Schrödinger equation. The laser field is described by a space- and time-dependent electric field. In order to solve numerically the Schrödinger equation, we modify the transfer-matrix algorithm such that it becomes numerically stable for a very large number of matching points. This modification enables one to investigate space effects related to a finite size of a laser focus. Numerical analysis shows the importance of finite penetration depth of laser fields in solids and the ponderomotive energy related to the quiver motion of electrons in the laser focus. We demonstrate that the energy spectrum of emitted electrons and the total photocurrent depend on the relative and carrier-envelope phases, indicating that multiphoton dynamics of electrons emitted from a solid surface can be efficiently controlled by varying these phases.

© 2011 Optical Society of America

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  1. N. Tzoar and J. I. Gersten, “Theory of electronic band structure in intense laser fields,” Phys. Rev. B 12, 1132–1139 (1975).
    [CrossRef]
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    [CrossRef]
  3. F. H. M. Faisal and R. Genieser, “Exact solution of the Kronig-Penney model of 1D-crystals in strong laser fields,” Phys. Lett. A 141, 297–300 (1989).
    [CrossRef]
  4. J. Z. Kamiński, “Laser-induced modification of the energy band structure,” Acta Phys. Pol. A 83, 495–504 (1993).
  5. D. R. Mašović, M. R. Belić, and J. I. Gersten, “Solid in the kicking laser field,” Phys. Lett. A 373, 3289–3295 (2009).
    [CrossRef]
  6. R. P. Lungu, “Floquet solutions for two-dimensional quasi-free electrons interacting with a monochromatic laser field,” Phys. Scr. 75, 206–215 (2007).
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  7. H. Khosravi, N. Daneshfar, and A. Bahari, “Effect of a magnetic field on high-harmonic generation by carbon nanotubes,” Opt. Lett. 34, 1723–1725 (2009).
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  8. H. Hsu and L. E. Reichl, “Floquet-Bloch states, quasienergy bands, and high-order harmonic generation for single-walled carbon nanotubes under intense laser fields,” Phys. Rev. B 74, 115406 (2006).
    [CrossRef]
  9. H. Hsu and L. E. Reichl, “Modeling graphene layers and single-walled carbon nanotubes with regularized delta-function potentials,” Phys. Rev. B 72, 155413 (2005).
    [CrossRef]
  10. M. Faraggi, I. Aldazabal, M. S. Gravielle, A. Arnau, and V. M. Silkin, “Study of the induced potential produced by ultrashort pulses on metal surfaces,” J. Opt. Soc. Am. B 26, 2331–2236 (2009).
    [CrossRef]
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    [CrossRef]
  12. J. C. Baggesen and L. B. Madsen, “Theory for time-resolved measurements of laser-induced electron emission from metal surfaces,” Phys. Rev. A 78, 032903 (2008).
    [CrossRef]
  13. M. N. Faraggi, M. S. Gravielle, and D. M. Mitnik, “Interaction of ultrashort laser pulses with metal surfaces: impulsive jellium-Volkov approximation versus the solution of the time-dependent Schrodinger equation,” Phys. Rev. A 76, 012903 (2007).
    [CrossRef]
  14. P. Dombi, F. Krausz, and G. Farkas, “Ultrafast dynamics and carrier-envelope phase sensitivity of multiphoton photoemission from metal surfaces,” J. Mod. Opt. 53, 163–172 (2006).
    [CrossRef]
  15. F. H. M. Faisal, J. Z. Kamiński, and E. Saczuk, “Photoemission and high-order harmonic generation from solid surfaces in intense laser fields,” Phys. Rev. A 72, 023412 (2005).
    [CrossRef]
  16. F. H. M. Faisal, J. Z. Kamiński, and E. Saczuk, “Photoemission from solid surfaces in intense laser fields,” Laser Phys. 16, 284–288 (2006).
    [CrossRef]
  17. V. A. Astapenko, “Radiative processes in a bichromatic laser field with multiple frequencies,” Quantum Electron. 36, 1131–1147 (2006).
    [CrossRef]
  18. F. Ehlotzky, “Atomic phenomena in bichromatic laser fields,” Phys. Rep. 345, 175–264 (2001).
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  21. E. Saczuk and J. Z. Kamiński, “Resonant tunneling in the presence of intense laser fields,” Phys. Status Solidi B 240, 603–609 (2003).
    [CrossRef]
  22. S. G. Davison and M. Ste¸ślicka, Basic Theory of Surface States (Clarendon, 2001).
  23. M. Shapiro and P. Brumer, “Laser control of product quantum state populations in unimolecular reactions,” J. Chem. Phys. 84, 4103–4104 (1986).
    [CrossRef]
  24. M. Shapiro and P. Brumer, “Coherent control of atomic molecular, and electronic processes,” Adv. At. Mol. Opt. Phys. 42, 287–345 (2000).
    [CrossRef]
  25. M. Shapiro and P. Brumer, “Coherent control of molecular dynamics,” Rep. Prog. Phys. 66, 859–942 (2003).
    [CrossRef]
  26. E. Moon, H. Wang, S. Gilbertson, H. Mashiko, M. Chini, and Z. Chang, “Advances in carrier-envelope phase stabilization of grating-based chirped-pulse amplifiers,” Laser Photon. Rev. 4, 160–177 (2010).
    [CrossRef]
  27. T. M. Fortier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, “Carrier-envelope phase-controlled quantum interference of injected photocurrents in semiconductors,” Phys. Rev. Lett. 92, 147403 (2004).
    [CrossRef] [PubMed]
  28. A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, “Observation of light-phase-sensitive photoemission from a metal,” Phys. Rev. Lett. 92, 073902 (2004).
    [CrossRef] [PubMed]
  29. S. E. Irvine, P. Dombi, G. Farkas, and A. Y. Elezzabi, “Influence of the carrier-envelope phase of few-cycle pulses on ponderomotive surface-plasmon electron acceleration,” Phys. Rev. Lett. 97, 146801 (2006).
    [CrossRef] [PubMed]
  30. A. T. Georges and N. E. Karatzas, “Modeling of ultrafast interferometric three-photon photoemission from a metal surface irradiated with sub-10 fs laser pulses,” Phys. Rev. B 77, 085436 (2008).
    [CrossRef]
  31. G. G. Paulus, F. Grasbon, H. Walther, P. Villoresi, M. Nisoli, S. Stagira, E. Priori, and S. De Silvestri, “Absolute-phase phenomena in photoionization with few-cycle laser pulses,” Nature 414, 182–184 (2001).
    [CrossRef] [PubMed]
  32. G. G. Paulus, F. Lindner, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004 (2003).
    [CrossRef]
  33. D. B. Milošević, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B 39, R203–R262 (2006).
    [CrossRef]

2010 (1)

E. Moon, H. Wang, S. Gilbertson, H. Mashiko, M. Chini, and Z. Chang, “Advances in carrier-envelope phase stabilization of grating-based chirped-pulse amplifiers,” Laser Photon. Rev. 4, 160–177 (2010).
[CrossRef]

2009 (3)

2008 (3)

A. T. Georges and N. E. Karatzas, “Modeling of ultrafast interferometric three-photon photoemission from a metal surface irradiated with sub-10 fs laser pulses,” Phys. Rev. B 77, 085436 (2008).
[CrossRef]

G. Saathoff, L. Miaja-Avila, M. Aeschlimann, M. M. Murname, and H. C. Kapteyn, “Laser-assisted photoemission from surfaces,” Phys. Rev. A 77, 022903 (2008).
[CrossRef]

J. C. Baggesen and L. B. Madsen, “Theory for time-resolved measurements of laser-induced electron emission from metal surfaces,” Phys. Rev. A 78, 032903 (2008).
[CrossRef]

2007 (2)

M. N. Faraggi, M. S. Gravielle, and D. M. Mitnik, “Interaction of ultrashort laser pulses with metal surfaces: impulsive jellium-Volkov approximation versus the solution of the time-dependent Schrodinger equation,” Phys. Rev. A 76, 012903 (2007).
[CrossRef]

R. P. Lungu, “Floquet solutions for two-dimensional quasi-free electrons interacting with a monochromatic laser field,” Phys. Scr. 75, 206–215 (2007).
[CrossRef]

2006 (6)

H. Hsu and L. E. Reichl, “Floquet-Bloch states, quasienergy bands, and high-order harmonic generation for single-walled carbon nanotubes under intense laser fields,” Phys. Rev. B 74, 115406 (2006).
[CrossRef]

S. E. Irvine, P. Dombi, G. Farkas, and A. Y. Elezzabi, “Influence of the carrier-envelope phase of few-cycle pulses on ponderomotive surface-plasmon electron acceleration,” Phys. Rev. Lett. 97, 146801 (2006).
[CrossRef] [PubMed]

D. B. Milošević, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B 39, R203–R262 (2006).
[CrossRef]

P. Dombi, F. Krausz, and G. Farkas, “Ultrafast dynamics and carrier-envelope phase sensitivity of multiphoton photoemission from metal surfaces,” J. Mod. Opt. 53, 163–172 (2006).
[CrossRef]

F. H. M. Faisal, J. Z. Kamiński, and E. Saczuk, “Photoemission from solid surfaces in intense laser fields,” Laser Phys. 16, 284–288 (2006).
[CrossRef]

V. A. Astapenko, “Radiative processes in a bichromatic laser field with multiple frequencies,” Quantum Electron. 36, 1131–1147 (2006).
[CrossRef]

2005 (2)

F. H. M. Faisal, J. Z. Kamiński, and E. Saczuk, “Photoemission and high-order harmonic generation from solid surfaces in intense laser fields,” Phys. Rev. A 72, 023412 (2005).
[CrossRef]

H. Hsu and L. E. Reichl, “Modeling graphene layers and single-walled carbon nanotubes with regularized delta-function potentials,” Phys. Rev. B 72, 155413 (2005).
[CrossRef]

2004 (2)

T. M. Fortier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, “Carrier-envelope phase-controlled quantum interference of injected photocurrents in semiconductors,” Phys. Rev. Lett. 92, 147403 (2004).
[CrossRef] [PubMed]

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, “Observation of light-phase-sensitive photoemission from a metal,” Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

2003 (3)

E. Saczuk and J. Z. Kamiński, “Resonant tunneling in the presence of intense laser fields,” Phys. Status Solidi B 240, 603–609 (2003).
[CrossRef]

M. Shapiro and P. Brumer, “Coherent control of molecular dynamics,” Rep. Prog. Phys. 66, 859–942 (2003).
[CrossRef]

G. G. Paulus, F. Lindner, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

2001 (2)

G. G. Paulus, F. Grasbon, H. Walther, P. Villoresi, M. Nisoli, S. Stagira, E. Priori, and S. De Silvestri, “Absolute-phase phenomena in photoionization with few-cycle laser pulses,” Nature 414, 182–184 (2001).
[CrossRef] [PubMed]

F. Ehlotzky, “Atomic phenomena in bichromatic laser fields,” Phys. Rep. 345, 175–264 (2001).
[CrossRef]

2000 (1)

M. Shapiro and P. Brumer, “Coherent control of atomic molecular, and electronic processes,” Adv. At. Mol. Opt. Phys. 42, 287–345 (2000).
[CrossRef]

1993 (1)

J. Z. Kamiński, “Laser-induced modification of the energy band structure,” Acta Phys. Pol. A 83, 495–504 (1993).

1989 (1)

F. H. M. Faisal and R. Genieser, “Exact solution of the Kronig-Penney model of 1D-crystals in strong laser fields,” Phys. Lett. A 141, 297–300 (1989).
[CrossRef]

1987 (1)

M. R. Belić, “Quasi-energy band-structure of solids,” Solid State Commun. 62, 817–820 (1987).
[CrossRef]

1986 (1)

M. Shapiro and P. Brumer, “Laser control of product quantum state populations in unimolecular reactions,” J. Chem. Phys. 84, 4103–4104 (1986).
[CrossRef]

1977 (1)

H. B. Michelson, “The work function of the elements and its periodicity,” J. Appl. Phys. 48, 4729–4733 (1977).
[CrossRef]

1975 (1)

N. Tzoar and J. I. Gersten, “Theory of electronic band structure in intense laser fields,” Phys. Rev. B 12, 1132–1139 (1975).
[CrossRef]

Aeschlimann, M.

G. Saathoff, L. Miaja-Avila, M. Aeschlimann, M. M. Murname, and H. C. Kapteyn, “Laser-assisted photoemission from surfaces,” Phys. Rev. A 77, 022903 (2008).
[CrossRef]

Aldazabal, I.

Apolonski, A.

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, “Observation of light-phase-sensitive photoemission from a metal,” Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

Arnau, A.

Astapenko, V. A.

V. A. Astapenko, “Radiative processes in a bichromatic laser field with multiple frequencies,” Quantum Electron. 36, 1131–1147 (2006).
[CrossRef]

Baggesen, J. C.

J. C. Baggesen and L. B. Madsen, “Theory for time-resolved measurements of laser-induced electron emission from metal surfaces,” Phys. Rev. A 78, 032903 (2008).
[CrossRef]

Bahari, A.

Baltuška, A.

G. G. Paulus, F. Lindner, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

Bauer, D.

D. B. Milošević, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B 39, R203–R262 (2006).
[CrossRef]

Becker, W.

D. B. Milošević, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B 39, R203–R262 (2006).
[CrossRef]

Belic, M. R.

D. R. Mašović, M. R. Belić, and J. I. Gersten, “Solid in the kicking laser field,” Phys. Lett. A 373, 3289–3295 (2009).
[CrossRef]

M. R. Belić, “Quasi-energy band-structure of solids,” Solid State Commun. 62, 817–820 (1987).
[CrossRef]

Bhat, R. D. R.

T. M. Fortier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, “Carrier-envelope phase-controlled quantum interference of injected photocurrents in semiconductors,” Phys. Rev. Lett. 92, 147403 (2004).
[CrossRef] [PubMed]

Brumer, P.

M. Shapiro and P. Brumer, “Coherent control of molecular dynamics,” Rep. Prog. Phys. 66, 859–942 (2003).
[CrossRef]

M. Shapiro and P. Brumer, “Coherent control of atomic molecular, and electronic processes,” Adv. At. Mol. Opt. Phys. 42, 287–345 (2000).
[CrossRef]

M. Shapiro and P. Brumer, “Laser control of product quantum state populations in unimolecular reactions,” J. Chem. Phys. 84, 4103–4104 (1986).
[CrossRef]

Burgdörfer, J.

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, “Observation of light-phase-sensitive photoemission from a metal,” Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

Chang, Z.

E. Moon, H. Wang, S. Gilbertson, H. Mashiko, M. Chini, and Z. Chang, “Advances in carrier-envelope phase stabilization of grating-based chirped-pulse amplifiers,” Laser Photon. Rev. 4, 160–177 (2010).
[CrossRef]

Chini, M.

E. Moon, H. Wang, S. Gilbertson, H. Mashiko, M. Chini, and Z. Chang, “Advances in carrier-envelope phase stabilization of grating-based chirped-pulse amplifiers,” Laser Photon. Rev. 4, 160–177 (2010).
[CrossRef]

Cundiff, S. T.

T. M. Fortier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, “Carrier-envelope phase-controlled quantum interference of injected photocurrents in semiconductors,” Phys. Rev. Lett. 92, 147403 (2004).
[CrossRef] [PubMed]

Daneshfar, N.

Davison, S. G.

S. G. Davison and M. Ste¸ślicka, Basic Theory of Surface States (Clarendon, 2001).

De Silvestri, S.

G. G. Paulus, F. Grasbon, H. Walther, P. Villoresi, M. Nisoli, S. Stagira, E. Priori, and S. De Silvestri, “Absolute-phase phenomena in photoionization with few-cycle laser pulses,” Nature 414, 182–184 (2001).
[CrossRef] [PubMed]

Dombi, P.

S. E. Irvine, P. Dombi, G. Farkas, and A. Y. Elezzabi, “Influence of the carrier-envelope phase of few-cycle pulses on ponderomotive surface-plasmon electron acceleration,” Phys. Rev. Lett. 97, 146801 (2006).
[CrossRef] [PubMed]

P. Dombi, F. Krausz, and G. Farkas, “Ultrafast dynamics and carrier-envelope phase sensitivity of multiphoton photoemission from metal surfaces,” J. Mod. Opt. 53, 163–172 (2006).
[CrossRef]

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, “Observation of light-phase-sensitive photoemission from a metal,” Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

Ehlotzky, F.

F. Ehlotzky, “Atomic phenomena in bichromatic laser fields,” Phys. Rep. 345, 175–264 (2001).
[CrossRef]

Elezzabi, A. Y.

S. E. Irvine, P. Dombi, G. Farkas, and A. Y. Elezzabi, “Influence of the carrier-envelope phase of few-cycle pulses on ponderomotive surface-plasmon electron acceleration,” Phys. Rev. Lett. 97, 146801 (2006).
[CrossRef] [PubMed]

Faisal, F. H. M.

F. H. M. Faisal, J. Z. Kamiński, and E. Saczuk, “Photoemission from solid surfaces in intense laser fields,” Laser Phys. 16, 284–288 (2006).
[CrossRef]

F. H. M. Faisal, J. Z. Kamiński, and E. Saczuk, “Photoemission and high-order harmonic generation from solid surfaces in intense laser fields,” Phys. Rev. A 72, 023412 (2005).
[CrossRef]

F. H. M. Faisal and R. Genieser, “Exact solution of the Kronig-Penney model of 1D-crystals in strong laser fields,” Phys. Lett. A 141, 297–300 (1989).
[CrossRef]

Faraggi, M.

Faraggi, M. N.

M. N. Faraggi, M. S. Gravielle, and D. M. Mitnik, “Interaction of ultrashort laser pulses with metal surfaces: impulsive jellium-Volkov approximation versus the solution of the time-dependent Schrodinger equation,” Phys. Rev. A 76, 012903 (2007).
[CrossRef]

Farkas, G.

P. Dombi, F. Krausz, and G. Farkas, “Ultrafast dynamics and carrier-envelope phase sensitivity of multiphoton photoemission from metal surfaces,” J. Mod. Opt. 53, 163–172 (2006).
[CrossRef]

S. E. Irvine, P. Dombi, G. Farkas, and A. Y. Elezzabi, “Influence of the carrier-envelope phase of few-cycle pulses on ponderomotive surface-plasmon electron acceleration,” Phys. Rev. Lett. 97, 146801 (2006).
[CrossRef] [PubMed]

Fortier, T. M.

T. M. Fortier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, “Carrier-envelope phase-controlled quantum interference of injected photocurrents in semiconductors,” Phys. Rev. Lett. 92, 147403 (2004).
[CrossRef] [PubMed]

Genieser, R.

F. H. M. Faisal and R. Genieser, “Exact solution of the Kronig-Penney model of 1D-crystals in strong laser fields,” Phys. Lett. A 141, 297–300 (1989).
[CrossRef]

Georges, A. T.

A. T. Georges and N. E. Karatzas, “Modeling of ultrafast interferometric three-photon photoemission from a metal surface irradiated with sub-10 fs laser pulses,” Phys. Rev. B 77, 085436 (2008).
[CrossRef]

Gersten, J. I.

D. R. Mašović, M. R. Belić, and J. I. Gersten, “Solid in the kicking laser field,” Phys. Lett. A 373, 3289–3295 (2009).
[CrossRef]

N. Tzoar and J. I. Gersten, “Theory of electronic band structure in intense laser fields,” Phys. Rev. B 12, 1132–1139 (1975).
[CrossRef]

Gilbertson, S.

E. Moon, H. Wang, S. Gilbertson, H. Mashiko, M. Chini, and Z. Chang, “Advances in carrier-envelope phase stabilization of grating-based chirped-pulse amplifiers,” Laser Photon. Rev. 4, 160–177 (2010).
[CrossRef]

Goulielmakis, E.

G. G. Paulus, F. Lindner, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

Grasbon, F.

G. G. Paulus, F. Grasbon, H. Walther, P. Villoresi, M. Nisoli, S. Stagira, E. Priori, and S. De Silvestri, “Absolute-phase phenomena in photoionization with few-cycle laser pulses,” Nature 414, 182–184 (2001).
[CrossRef] [PubMed]

Gravielle, M. S.

M. Faraggi, I. Aldazabal, M. S. Gravielle, A. Arnau, and V. M. Silkin, “Study of the induced potential produced by ultrashort pulses on metal surfaces,” J. Opt. Soc. Am. B 26, 2331–2236 (2009).
[CrossRef]

M. N. Faraggi, M. S. Gravielle, and D. M. Mitnik, “Interaction of ultrashort laser pulses with metal surfaces: impulsive jellium-Volkov approximation versus the solution of the time-dependent Schrodinger equation,” Phys. Rev. A 76, 012903 (2007).
[CrossRef]

Hänsch, T. W.

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, “Observation of light-phase-sensitive photoemission from a metal,” Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

Holzwarth, R.

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, “Observation of light-phase-sensitive photoemission from a metal,” Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

Hsu, H.

H. Hsu and L. E. Reichl, “Floquet-Bloch states, quasienergy bands, and high-order harmonic generation for single-walled carbon nanotubes under intense laser fields,” Phys. Rev. B 74, 115406 (2006).
[CrossRef]

H. Hsu and L. E. Reichl, “Modeling graphene layers and single-walled carbon nanotubes with regularized delta-function potentials,” Phys. Rev. B 72, 155413 (2005).
[CrossRef]

Irvine, S. E.

S. E. Irvine, P. Dombi, G. Farkas, and A. Y. Elezzabi, “Influence of the carrier-envelope phase of few-cycle pulses on ponderomotive surface-plasmon electron acceleration,” Phys. Rev. Lett. 97, 146801 (2006).
[CrossRef] [PubMed]

Jones, D. J.

T. M. Fortier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, “Carrier-envelope phase-controlled quantum interference of injected photocurrents in semiconductors,” Phys. Rev. Lett. 92, 147403 (2004).
[CrossRef] [PubMed]

Kakehata, M.

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, “Observation of light-phase-sensitive photoemission from a metal,” Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

Kaminski, J. Z.

F. H. M. Faisal, J. Z. Kamiński, and E. Saczuk, “Photoemission from solid surfaces in intense laser fields,” Laser Phys. 16, 284–288 (2006).
[CrossRef]

F. H. M. Faisal, J. Z. Kamiński, and E. Saczuk, “Photoemission and high-order harmonic generation from solid surfaces in intense laser fields,” Phys. Rev. A 72, 023412 (2005).
[CrossRef]

E. Saczuk and J. Z. Kamiński, “Resonant tunneling in the presence of intense laser fields,” Phys. Status Solidi B 240, 603–609 (2003).
[CrossRef]

J. Z. Kamiński, “Laser-induced modification of the energy band structure,” Acta Phys. Pol. A 83, 495–504 (1993).

Kapteyn, H. C.

G. Saathoff, L. Miaja-Avila, M. Aeschlimann, M. M. Murname, and H. C. Kapteyn, “Laser-assisted photoemission from surfaces,” Phys. Rev. A 77, 022903 (2008).
[CrossRef]

Karatzas, N. E.

A. T. Georges and N. E. Karatzas, “Modeling of ultrafast interferometric three-photon photoemission from a metal surface irradiated with sub-10 fs laser pulses,” Phys. Rev. B 77, 085436 (2008).
[CrossRef]

Khosravi, H.

Krausz, F.

P. Dombi, F. Krausz, and G. Farkas, “Ultrafast dynamics and carrier-envelope phase sensitivity of multiphoton photoemission from metal surfaces,” J. Mod. Opt. 53, 163–172 (2006).
[CrossRef]

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, “Observation of light-phase-sensitive photoemission from a metal,” Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

G. G. Paulus, F. Lindner, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

Lemell, C.

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, “Observation of light-phase-sensitive photoemission from a metal,” Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

Lezius, M.

G. G. Paulus, F. Lindner, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

Lindner, F.

G. G. Paulus, F. Lindner, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

Lungu, R. P.

R. P. Lungu, “Floquet solutions for two-dimensional quasi-free electrons interacting with a monochromatic laser field,” Phys. Scr. 75, 206–215 (2007).
[CrossRef]

Madsen, L. B.

J. C. Baggesen and L. B. Madsen, “Theory for time-resolved measurements of laser-induced electron emission from metal surfaces,” Phys. Rev. A 78, 032903 (2008).
[CrossRef]

Mashiko, H.

E. Moon, H. Wang, S. Gilbertson, H. Mashiko, M. Chini, and Z. Chang, “Advances in carrier-envelope phase stabilization of grating-based chirped-pulse amplifiers,” Laser Photon. Rev. 4, 160–177 (2010).
[CrossRef]

Mašovic, D. R.

D. R. Mašović, M. R. Belić, and J. I. Gersten, “Solid in the kicking laser field,” Phys. Lett. A 373, 3289–3295 (2009).
[CrossRef]

Miaja-Avila, L.

G. Saathoff, L. Miaja-Avila, M. Aeschlimann, M. M. Murname, and H. C. Kapteyn, “Laser-assisted photoemission from surfaces,” Phys. Rev. A 77, 022903 (2008).
[CrossRef]

Michelson, H. B.

H. B. Michelson, “The work function of the elements and its periodicity,” J. Appl. Phys. 48, 4729–4733 (1977).
[CrossRef]

Miloševic, D. B.

D. B. Milošević, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B 39, R203–R262 (2006).
[CrossRef]

Mitnik, D. M.

M. N. Faraggi, M. S. Gravielle, and D. M. Mitnik, “Interaction of ultrashort laser pulses with metal surfaces: impulsive jellium-Volkov approximation versus the solution of the time-dependent Schrodinger equation,” Phys. Rev. A 76, 012903 (2007).
[CrossRef]

Mizutani, U.

U. Mizutani, Introduction to the Electron Theory of Metals (Cambridge University, 2001).
[CrossRef]

Moon, E.

E. Moon, H. Wang, S. Gilbertson, H. Mashiko, M. Chini, and Z. Chang, “Advances in carrier-envelope phase stabilization of grating-based chirped-pulse amplifiers,” Laser Photon. Rev. 4, 160–177 (2010).
[CrossRef]

Murname, M. M.

G. Saathoff, L. Miaja-Avila, M. Aeschlimann, M. M. Murname, and H. C. Kapteyn, “Laser-assisted photoemission from surfaces,” Phys. Rev. A 77, 022903 (2008).
[CrossRef]

Nisoli, M.

G. G. Paulus, F. Grasbon, H. Walther, P. Villoresi, M. Nisoli, S. Stagira, E. Priori, and S. De Silvestri, “Absolute-phase phenomena in photoionization with few-cycle laser pulses,” Nature 414, 182–184 (2001).
[CrossRef] [PubMed]

Paulus, G. G.

D. B. Milošević, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B 39, R203–R262 (2006).
[CrossRef]

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, “Observation of light-phase-sensitive photoemission from a metal,” Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

G. G. Paulus, F. Lindner, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

G. G. Paulus, F. Grasbon, H. Walther, P. Villoresi, M. Nisoli, S. Stagira, E. Priori, and S. De Silvestri, “Absolute-phase phenomena in photoionization with few-cycle laser pulses,” Nature 414, 182–184 (2001).
[CrossRef] [PubMed]

Priori, E.

G. G. Paulus, F. Grasbon, H. Walther, P. Villoresi, M. Nisoli, S. Stagira, E. Priori, and S. De Silvestri, “Absolute-phase phenomena in photoionization with few-cycle laser pulses,” Nature 414, 182–184 (2001).
[CrossRef] [PubMed]

Reichl, L. E.

H. Hsu and L. E. Reichl, “Floquet-Bloch states, quasienergy bands, and high-order harmonic generation for single-walled carbon nanotubes under intense laser fields,” Phys. Rev. B 74, 115406 (2006).
[CrossRef]

H. Hsu and L. E. Reichl, “Modeling graphene layers and single-walled carbon nanotubes with regularized delta-function potentials,” Phys. Rev. B 72, 155413 (2005).
[CrossRef]

Roos, P. A.

T. M. Fortier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, “Carrier-envelope phase-controlled quantum interference of injected photocurrents in semiconductors,” Phys. Rev. Lett. 92, 147403 (2004).
[CrossRef] [PubMed]

Saathoff, G.

G. Saathoff, L. Miaja-Avila, M. Aeschlimann, M. M. Murname, and H. C. Kapteyn, “Laser-assisted photoemission from surfaces,” Phys. Rev. A 77, 022903 (2008).
[CrossRef]

Saczuk, E.

F. H. M. Faisal, J. Z. Kamiński, and E. Saczuk, “Photoemission from solid surfaces in intense laser fields,” Laser Phys. 16, 284–288 (2006).
[CrossRef]

F. H. M. Faisal, J. Z. Kamiński, and E. Saczuk, “Photoemission and high-order harmonic generation from solid surfaces in intense laser fields,” Phys. Rev. A 72, 023412 (2005).
[CrossRef]

E. Saczuk and J. Z. Kamiński, “Resonant tunneling in the presence of intense laser fields,” Phys. Status Solidi B 240, 603–609 (2003).
[CrossRef]

Shapiro, M.

M. Shapiro and P. Brumer, “Coherent control of molecular dynamics,” Rep. Prog. Phys. 66, 859–942 (2003).
[CrossRef]

M. Shapiro and P. Brumer, “Coherent control of atomic molecular, and electronic processes,” Adv. At. Mol. Opt. Phys. 42, 287–345 (2000).
[CrossRef]

M. Shapiro and P. Brumer, “Laser control of product quantum state populations in unimolecular reactions,” J. Chem. Phys. 84, 4103–4104 (1986).
[CrossRef]

Silkin, V. M.

Sipe, J. E.

T. M. Fortier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, “Carrier-envelope phase-controlled quantum interference of injected photocurrents in semiconductors,” Phys. Rev. Lett. 92, 147403 (2004).
[CrossRef] [PubMed]

Stagira, S.

G. G. Paulus, F. Grasbon, H. Walther, P. Villoresi, M. Nisoli, S. Stagira, E. Priori, and S. De Silvestri, “Absolute-phase phenomena in photoionization with few-cycle laser pulses,” Nature 414, 182–184 (2001).
[CrossRef] [PubMed]

Ste¸slicka, M.

S. G. Davison and M. Ste¸ślicka, Basic Theory of Surface States (Clarendon, 2001).

Torizuka, K.

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, “Observation of light-phase-sensitive photoemission from a metal,” Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

Tzoar, N.

N. Tzoar and J. I. Gersten, “Theory of electronic band structure in intense laser fields,” Phys. Rev. B 12, 1132–1139 (1975).
[CrossRef]

Udem, T.

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, “Observation of light-phase-sensitive photoemission from a metal,” Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

Villoresi, P.

G. G. Paulus, F. Grasbon, H. Walther, P. Villoresi, M. Nisoli, S. Stagira, E. Priori, and S. De Silvestri, “Absolute-phase phenomena in photoionization with few-cycle laser pulses,” Nature 414, 182–184 (2001).
[CrossRef] [PubMed]

Walther, H.

G. G. Paulus, F. Lindner, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

G. G. Paulus, F. Grasbon, H. Walther, P. Villoresi, M. Nisoli, S. Stagira, E. Priori, and S. De Silvestri, “Absolute-phase phenomena in photoionization with few-cycle laser pulses,” Nature 414, 182–184 (2001).
[CrossRef] [PubMed]

Wang, H.

E. Moon, H. Wang, S. Gilbertson, H. Mashiko, M. Chini, and Z. Chang, “Advances in carrier-envelope phase stabilization of grating-based chirped-pulse amplifiers,” Laser Photon. Rev. 4, 160–177 (2010).
[CrossRef]

Acta Phys. Pol. A (1)

J. Z. Kamiński, “Laser-induced modification of the energy band structure,” Acta Phys. Pol. A 83, 495–504 (1993).

Adv. At. Mol. Opt. Phys. (1)

M. Shapiro and P. Brumer, “Coherent control of atomic molecular, and electronic processes,” Adv. At. Mol. Opt. Phys. 42, 287–345 (2000).
[CrossRef]

J. Appl. Phys. (1)

H. B. Michelson, “The work function of the elements and its periodicity,” J. Appl. Phys. 48, 4729–4733 (1977).
[CrossRef]

J. Chem. Phys. (1)

M. Shapiro and P. Brumer, “Laser control of product quantum state populations in unimolecular reactions,” J. Chem. Phys. 84, 4103–4104 (1986).
[CrossRef]

J. Mod. Opt. (1)

P. Dombi, F. Krausz, and G. Farkas, “Ultrafast dynamics and carrier-envelope phase sensitivity of multiphoton photoemission from metal surfaces,” J. Mod. Opt. 53, 163–172 (2006).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. B (1)

D. B. Milošević, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B 39, R203–R262 (2006).
[CrossRef]

Laser Photon. Rev. (1)

E. Moon, H. Wang, S. Gilbertson, H. Mashiko, M. Chini, and Z. Chang, “Advances in carrier-envelope phase stabilization of grating-based chirped-pulse amplifiers,” Laser Photon. Rev. 4, 160–177 (2010).
[CrossRef]

Laser Phys. (1)

F. H. M. Faisal, J. Z. Kamiński, and E. Saczuk, “Photoemission from solid surfaces in intense laser fields,” Laser Phys. 16, 284–288 (2006).
[CrossRef]

Nature (1)

G. G. Paulus, F. Grasbon, H. Walther, P. Villoresi, M. Nisoli, S. Stagira, E. Priori, and S. De Silvestri, “Absolute-phase phenomena in photoionization with few-cycle laser pulses,” Nature 414, 182–184 (2001).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Lett. A (2)

F. H. M. Faisal and R. Genieser, “Exact solution of the Kronig-Penney model of 1D-crystals in strong laser fields,” Phys. Lett. A 141, 297–300 (1989).
[CrossRef]

D. R. Mašović, M. R. Belić, and J. I. Gersten, “Solid in the kicking laser field,” Phys. Lett. A 373, 3289–3295 (2009).
[CrossRef]

Phys. Rep. (1)

F. Ehlotzky, “Atomic phenomena in bichromatic laser fields,” Phys. Rep. 345, 175–264 (2001).
[CrossRef]

Phys. Rev. A (4)

F. H. M. Faisal, J. Z. Kamiński, and E. Saczuk, “Photoemission and high-order harmonic generation from solid surfaces in intense laser fields,” Phys. Rev. A 72, 023412 (2005).
[CrossRef]

G. Saathoff, L. Miaja-Avila, M. Aeschlimann, M. M. Murname, and H. C. Kapteyn, “Laser-assisted photoemission from surfaces,” Phys. Rev. A 77, 022903 (2008).
[CrossRef]

J. C. Baggesen and L. B. Madsen, “Theory for time-resolved measurements of laser-induced electron emission from metal surfaces,” Phys. Rev. A 78, 032903 (2008).
[CrossRef]

M. N. Faraggi, M. S. Gravielle, and D. M. Mitnik, “Interaction of ultrashort laser pulses with metal surfaces: impulsive jellium-Volkov approximation versus the solution of the time-dependent Schrodinger equation,” Phys. Rev. A 76, 012903 (2007).
[CrossRef]

Phys. Rev. B (4)

N. Tzoar and J. I. Gersten, “Theory of electronic band structure in intense laser fields,” Phys. Rev. B 12, 1132–1139 (1975).
[CrossRef]

A. T. Georges and N. E. Karatzas, “Modeling of ultrafast interferometric three-photon photoemission from a metal surface irradiated with sub-10 fs laser pulses,” Phys. Rev. B 77, 085436 (2008).
[CrossRef]

H. Hsu and L. E. Reichl, “Floquet-Bloch states, quasienergy bands, and high-order harmonic generation for single-walled carbon nanotubes under intense laser fields,” Phys. Rev. B 74, 115406 (2006).
[CrossRef]

H. Hsu and L. E. Reichl, “Modeling graphene layers and single-walled carbon nanotubes with regularized delta-function potentials,” Phys. Rev. B 72, 155413 (2005).
[CrossRef]

Phys. Rev. Lett. (4)

G. G. Paulus, F. Lindner, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

T. M. Fortier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, “Carrier-envelope phase-controlled quantum interference of injected photocurrents in semiconductors,” Phys. Rev. Lett. 92, 147403 (2004).
[CrossRef] [PubMed]

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, “Observation of light-phase-sensitive photoemission from a metal,” Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

S. E. Irvine, P. Dombi, G. Farkas, and A. Y. Elezzabi, “Influence of the carrier-envelope phase of few-cycle pulses on ponderomotive surface-plasmon electron acceleration,” Phys. Rev. Lett. 97, 146801 (2006).
[CrossRef] [PubMed]

Phys. Scr. (1)

R. P. Lungu, “Floquet solutions for two-dimensional quasi-free electrons interacting with a monochromatic laser field,” Phys. Scr. 75, 206–215 (2007).
[CrossRef]

Phys. Status Solidi B (1)

E. Saczuk and J. Z. Kamiński, “Resonant tunneling in the presence of intense laser fields,” Phys. Status Solidi B 240, 603–609 (2003).
[CrossRef]

Quantum Electron. (1)

V. A. Astapenko, “Radiative processes in a bichromatic laser field with multiple frequencies,” Quantum Electron. 36, 1131–1147 (2006).
[CrossRef]

Rep. Prog. Phys. (1)

M. Shapiro and P. Brumer, “Coherent control of molecular dynamics,” Rep. Prog. Phys. 66, 859–942 (2003).
[CrossRef]

Solid State Commun. (1)

M. R. Belić, “Quasi-energy band-structure of solids,” Solid State Commun. 62, 817–820 (1987).
[CrossRef]

Other (2)

S. G. Davison and M. Ste¸ślicka, Basic Theory of Surface States (Clarendon, 2001).

U. Mizutani, Introduction to the Electron Theory of Metals (Cambridge University, 2001).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

V ( x ) , Eq. (1), for some particular values of parameters: w 0 = 5 , x 0 = 50 , σ = 9 (all in atomic units of length), v 0 = 1 , and V 0 = 6.05 eV . Red horizontal lines mark positions of four resonant states, the real parts of which are equal to E 1 = 1.58 ω , E 2 = 2.68 ω , E 3 = 3.4 ω , and E 4 = 3.84 ω for ω = 1.5498 eV . The increasing width of these resonances with the increasing energy is indicated symbolically by the widths of these lines.

Fig. 2
Fig. 2

Energy spectrum S ( E kin ) (in atomic units), Eq. (33), for the bichromatic laser fields of frequencies ( ω , 2 ω ) and ϵ = 2 (upper frame) and ( ω , 3 ω ) and ϵ = 3 (lower frame), and for the relative phase φ = 0 . Other laser field parameters are ω = 1.5498 eV and ξ = 0.1 . As indicated, the top curves correspond to case 1 in Eq. (6), whereas the green dotted–dashed and red curves correspond to the cases 2 and 3, respectively.

Fig. 3
Fig. 3

Total photocurrent J (in atomic units), Eq. (32), for the bichromatic laser fields of frequencies ( ω , 2 ω ) and ϵ = 2 (upper frame) and ( ω , 3 ω ) and ϵ = 3 (lower frame), and for the relative phases φ = π / 8 , = 0 , 1 , , 16 (points are connected by lines to guide the eye). Other laser field parameters are the same as in Fig. 2, but only the physically meaningful case 3 is presented.

Fig. 4
Fig. 4

Total photocurrent J (in atomic units), Eq. (35), for the bichromatic laser fields of frequencies ( ω , 3 ω ) and the electric field defined by Eq. (34) (upper frame) and its high-energy part J ( E ) for E = 8 ω (lower frame) for the carrier-envelope phases φ = π / 8 , = 0 , 1 , , 8 (points are connected by lines to guide the eye). Other laser field parameters are ξ = 0.1 and ω = 1.5498 eV .

Equations (35)

Equations on this page are rendered with MathJax. Learn more.

V ( x ) = V 0 f P ( x ) ,
f P ( x ) = g ( x w 0 ) + v 0 exp ( ( x x 0 ) 2 σ 2 ) ,
g ( x ) = 1 / ( 1 + e x ) .
E ( x , t ) = E 0 f L ( x ) ( sin ( ω t ) + ϵ sin ( n ω t + φ ) )
A ( x , t ) = E 0 ω f L ( x ) ( cos ( ω t ) + ϵ n cos ( n ω t + φ ) ) ,
f L ( x ) = { 1 , case 1 , g ( x ζ a ) case 2 , g ( x ζ a ) g ( b x μ ) case 3 ,
I ( x ) exp ( 2 x ζ )
i t ψ ( x , t ) = [ 1 2 ( 1 i x + A ( x , t ) ) 2 + V ( x ) ] ψ ( x , t ) .
ψ i ( x , t ) = M = exp ( i ( E + M ω ) t ) × σ = ± N = C i N σ B M N ( σ p i N ) exp ( i σ p i N x ) ,
p i N = 2 ( E + N ω V i U i ) ,
exp ( i Φ i N σ ( t ) ) = M = exp ( i M ω t ) B M N ( σ p i N )
Φ i N σ ( t ) = 0 t ( σ A i ( t ) p i N 1 2 ( A i 2 ( t ) A i 2 ( t ) ) ) d t .
C i = ( C i + C i ) , C i σ = ( C i , 1 σ C i , 0 σ C i , 1 σ ) ,
C j = ( C j + C j ) = ( T j i ++ T j i + T j i + T j i −- ) ( C i + C i ) = T j i C i ,
T j i = T j k T k i ,
S j i = ( S j i ++ S j i + S j i + S j i −- )
S j i ++ = ( T j i ) 1 T j i + , S j i + = ( T j i −- ) 1 , S j i + = T j i ++ T j i + ( T j i −- ) 1 T j i + , S j i −- = T j i + ( T j i −- ) 1 .
S j i ++ = S k i ++ + S k i + ( 1 S j k ++ S k i −- ) 1 S j k ++ S k i + , S j i + = S k i + ( 1 S j k ++ S k , i −- ) 1 S j , k + , S j i + = S j k + ( 1 S j k ++ S k i −- ) 1 S k i + , S j i −- = S j k −- + S j k + S k i −- ( 1 S j k ++ S k i −- ) 1 S j k + .
N N ref p N p 0 | R N | 2 + N N tr q N p 0 | T N | 2 = 1 ,
p N = 2 ( E + N ω U C )
q N = 2 ( E + N ω V 0 U V ) ,
P R ( N ) = p N p 0 | R N | 2 ,
P T ( N ) = q N p 0 | T N | 2
J inc = p 0 ,
J ref = N N ref p N | R N | 2 ,
J tr = N N tr q N | T N | 2 .
J inc = J ref + J tr .
J = 2 2 π E p E F , p > 0 d p J tr ,
E kin = q N 2 2 = p 2 2 V 0 + U C U V + N ω ,
J tr = 0 d E kin P ( E kin , p )
P ( E kin , p ) = N N tr q N | T N | 2 × δ ( p 2 2 V 0 + U C U V + N ω E kin ) .
J = 0 d E kin S ( E kin )
S ( E kin ) = 2 2 π E p E F , p > 0 d p P ( E kin , p ) .
E ( x , t ) = E 0 f L ( x ) ( sin ( n ω t + φ ) + sin ( ( n + 2 ) ω t + φ ) ) = 2 E 0 f L ( x ) cos ( ω t ) sin ( ( n + 1 ) ω t + φ ) .
J ( E ) = E d E kin S ( E kin ) ,

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